your's ain't golf. No modules, lest I would say

use Junk;do
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- 11 chars.. but that could be golfed down to use J;d - 7 chars ;-)

blokhead's solutions look neat, but they hog my cpu... (with 2355/12344 - still not finished in 3/4 hour ;-)

#!/usr/bin/perl -l
pop=~m|/|;($f,$g)=($`,$');sub d{int($_[1]/$_[0]+1)}sub
g{($x,$y)=@_;($x,$y)=($y,$x% $y)while$y;$x} sub re{($p
,$e,$r,$l)=@_;($p,$l)=($p*$l-$e*$r,$e*$l);$g=g($p,$l);
for($p,$l){$_/=$g};($p,$l)}while($f>1){push@o,"1/".d(#
$f,$g);($f,$g)=re($f,$g,1,d($f,$g));}print join' + ',#
@o,"$f/$g";# ungolfed and thus not for production use!
[download]

way too long...

<update> golfed down a bit... (198 chars, counting newlines).

($z,$n)=($_=pop)=~/(.+)\/(.+)/;$s='==';for(;;){$m=int($n/$z+1);$_
.=" $s 1/".($z==1?$n:$m);$z<=1&&last;($z,$n)=($z*$m-$n,$m*$n);($x
,$y)=($z,$n);($x,$y)=($y,$x%$y)while$y;$z/=$x;$n/=$x;$s='+'}print
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qwurx [shmem] ~> perl -l egy.pl 2355/12344
2355/12344 == 1/6 + 1/42 + 1/3282 + 1/15755059 + 1/744665636525384
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in no time... </update>

--shmem

_($_=" "x(1<<5)."?\n".q·/)Oo.  G°\        /
                              /\_¯/(q    /
----------------------------  \__(m.====·.(_("always off the crowd"))."·
");sub _{s./.($e="'Itrs `mnsgdq Gdbj O`qkdq")=~y/"-y/#-z/;$e.e && print}